Let us prove part (REF ). Denote temporarily by \(\mathcal {J}^{comb}_{g,n}\)  the cone stack associated to the combinatorial cone stack (REF ) and let
\(\mathcal {J}^{comb}_{g,n}(\operatorname{\bf {RPC}})\)  be its restriction to \(\operatorname{\bf {RPC}}\) .
According to [1]}, an object of \(\mathcal {J}^{comb}_{g,n}(\operatorname{\bf {RPC}})\)  is a triple \((G,D,l:\sigma \rightarrow \mathbb {R}_{\ge 0}^{E(G)})\) , where \((G,D)\in \operatorname{\mathcal {QD}iv}_{g,n}\)  and \(l:\sigma \rightarrow \mathbb {R}_{\ge 0}^{E(G)}\)  is a morphism in \(\operatorname{\bf {RPC}}\)  whose image is not contained in any proper face of \(\mathbb {R}_{\ge 0}^{E(G)}\) . By composing with the projections along the coordinates, the morphism \(l\)  is equivalent to the datum of a collection of surjective morphisms \(\lbrace l_e:\sigma \rightarrow \mathbb {R}_{\ge 0}\rbrace _{e\in E(G)}\)  in \(\operatorname{\bf {RPC}}\) . By passing to the toric monoids, this is equivalent to giving a collection of injective morphisms of monoids \(\lbrace l_e^*:\mathbb {N}\rightarrow S_{\sigma }\rbrace _{e\in E(G)}\) , which is indeed determined by the collection of non-zero elements \(\lbrace l_e^*(1)\in S_{\sigma }\rbrace _{e\in E(G)}\) . Therefore, the triple \((G,D,l:\sigma \rightarrow \mathbb {R}_{\ge 0}^{E(G)})\)  gives rise to (and it is completely determined by) an object \((\Gamma /\sigma , D)\)  of \(\mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\)  such that \(\Gamma /\sigma \)  is the \(n\) -marked genus \(g\)  quasi-stable tropical curve whose underlying graph is \((\Gamma ):=G\)  and whose metric is given by \(d_{\Gamma }(e):=l_e^*(1) \in S_{\sigma }\)  for any \(e\in E(G)\) .
Similarly, a morphism \((G,D,l:\sigma \rightarrow \mathbb {R}_{\ge 0}^{E(G)})\rightarrow (G^{\prime },D^{\prime },l^{\prime }:\sigma ^{\prime }\rightarrow \mathbb {R}_{\ge 0}^{E(G^{\prime })})\)  in \(\mathcal {J}^{comb}_{g,n}(\operatorname{\bf {RPC}})\)  is the datum of a morphism of cones \(f:\sigma \rightarrow \sigma ^{\prime }\) 
and a morphism \(\pi :(G^{\prime },D^{\prime })\rightarrow (G,D)\)  in \(\operatorname{\mathcal {QD}iv}_{g,n}\)  such that the following diagram commutes
\({\sigma [r]^{l} [d]_f & \mathbb {R}_{\ge 0}^{E(G)} @{^{(}->}[d]\\\sigma ^{\prime } [r]^{l^{\prime }} & \mathbb {R}_{\ge 0}^{E(G^{\prime })}}\) 
where the left arrow is induced by the inclusion \(\pi _E^*:E(G)\hookrightarrow E(G^{\prime })\)  that identifies the edges of \(G\)  with the edges of \(G^{\prime }\)  that are not contracted by \(\pi \) .
In terms of the objects \((\Gamma /\sigma , D)\)  and \((\Gamma ^{\prime }/\sigma ^{\prime }, D^{\prime })\)  of \(\mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\)  associated to, respectively, \((G,D,l:\sigma \rightarrow \mathbb {R}_{\ge 0}^{E(G)})\)  and \((G^{\prime },D^{\prime },l^{\prime }:\sigma ^{\prime }\rightarrow \mathbb {R}_{\ge 0}^{E(G^{\prime })})\)  as explained above, the commutativity of the above diagram means exactly that \(\pi \)  is compatible with \(f\)  in the sense of Definition REF ,
and therefore we get a (uniquely defined) morphism \((f,\pi ):(\Gamma /\sigma , D)\rightarrow (\Gamma ^{\prime }/\sigma ^{\prime }, D^{\prime })\)  in \(\mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\) .
The above discussion shows that we have an isomorphism \(\mathcal {J}^{comb}_{g,n}(\operatorname{\bf {RPC}})\cong \mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\)  of categories fibered in groupoids over \(\operatorname{\bf {RPC}}\) , which then gives rise to an isomorphism of stacks \(\mathcal {J}^{comb}_{g,n}\cong \mathcal {J}^{trop}_{g,n}\)  by [1]}.
Let us now prove part (REF ). We will temporarily denote by \(\mathcal {M}^{comb}_{g,n}\)  the cone stack associated to the combinatorial cone stack (REF ),
by \(\mathcal {M}^{comb}_{g,n}(\operatorname{\bf {RPC}})\)  its restriction to \(\operatorname{\bf {RPC}}\)  and by \(\Phi ^{\rm comb}:\mathcal {J}^{comb}_{g,n}\rightarrow \mathcal {M}^{comb}_{g,n}\)  the morphism of cone stacks induced by the morphism of combinatorial cone stacks
described in part (REF ). By definition, the morphism \(\Phi ^{\rm comb}\)  is given on objects by
\(\begin{split}\mathcal {J}^{comb}(\operatorname{\bf {RPC}})&\longrightarrow \mathcal {M}^{comb}_{g,n}(\operatorname{\bf {RPC}})\\ \big (G,D,l\colon \sigma \rightarrow \mathbb {R}_{\ge 0}^{E(G)}\big ) &\longmapsto \big (G^{\operatorname{st}}, l^{\operatorname{st}}\colon \sigma \xrightarrow{} \mathbb {R}_{\ge 0}^{E(G)}\xrightarrow{} \mathbb {R}_{\ge 0}^{E(G^{\operatorname{st}})}\big ).\end{split}\) 
In terms of the isomorphisms \(\mathcal {J}^{comb}_{g,n}\cong \mathcal {J}^{trop}_{g,n}\)  and \(\mathcal {M}^{comb}_{g,n}\cong \mathcal {M}^{trop}_{g,n}\) , this corresponds to sending an object \((\Gamma /\sigma ,D)\in \mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\)  into \(\Gamma ^{\operatorname{st}}/\sigma \in \mathcal {M}^{trop}_{g,n}(\operatorname{\bf {RPC}})\) . Similarly, a morphism \((f,\pi ): (G,D,l:\sigma \rightarrow \mathbb {R}_{\ge 0}^{E(G)})\rightarrow (G^{\prime },D^{\prime },l^{\prime }:\sigma ^{\prime }\rightarrow \mathbb {R}_{\ge 0}^{E(G^{\prime })})\)  of \(\mathcal {J}^{comb}_{g,n}(\operatorname{\bf {RPC}})\)  is sent via \(\Phi ^{\rm comb}\)  into the morphism \((f,\pi ^{\operatorname{st}}): (G^{\operatorname{st}},l^{\operatorname{st}}:\sigma \rightarrow \mathbb {R}_{\ge 0}^{E(G^{\operatorname{st}})})\rightarrow (G^{\prime \operatorname{st}},l^{\prime \operatorname{st}}:\sigma ^{\prime }\rightarrow \mathbb {R}_{\ge 0}^{E(G^{\prime \operatorname{st}})})\)  of \(\mathcal {M}^{comb}_{g,n}(\operatorname{\bf {RPC}})\) .
In terms of the isomorphisms \(\mathcal {J}^{comb}_{g,n}\cong \mathcal {J}^{trop}_{g,n}\)  and \(\mathcal {M}^{comb}_{g,n}\cong \mathcal {M}^{trop}_{g,n}\) , this corresponds to sending a morphism \((f,\pi ):(\Gamma /\sigma ,D)\rightarrow (\Gamma ^{\prime }/\sigma ^{\prime },D^{\prime })\)  in \(\mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\)  into the morphism \((f,\pi ^{\operatorname{st}}):\Gamma ^{\operatorname{st}}/\sigma \rightarrow \Gamma ^{\prime \operatorname{st}}/\sigma ^{\prime }\)  in \(\mathcal {M}^{trop}_{g,n}\) .
The above discussion shows that, under the isomorphisms \(\mathcal {J}^{comb}_{g,n}\cong \mathcal {J}^{trop}_{g,n}\)  and \(\mathcal {M}^{comb}_{g,n}\cong \mathcal {M}^{trop}_{g,n}\) , the two morphisms \(\Phi ^{\operatorname{trop}}\)  and \(\Phi ^{\rm comb}\)  coincide when restricted to \(\operatorname{\bf {RPC}}\) , and hence they coincide everywhere by [1]}.
